Why do physics professionals often use various different systems of units instead of SI units. Especially I ask about when constants like $c$ or $\hbar$ are put to 1....what is the advantage of this?
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Everything!!! Natural units just make your life simpler, is like removing every unnecessary conversion factors which need to be added, helps by making the equation look smaller, and can easily give back the final equation just by dimensional analysis.
Okay you might be wondering what dimensional analysis has got to do with natural units. Well let's see an example. The Lorentz transformation factor is given as $$\gamma =\frac{1}{\sqrt{1-v^2}}$$ Now obviously, this equation needs to be converted back to some units that people are familiar with to make sense. So you know that the speed of light in natural units is 1, so you can multiply it anywhere, with any exponent. So if you analyse the square root, it must be unitless, but 1-v² is definitely not unitless soyou make the v² as v²/c², and behold, you get the Lorentz transformation factor as
$$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
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